Concentrations of chemicals in river water provide crucial information for assessing environmental exposure to fertilisers and insecticides, heavy metals, illicit drugs, pathogens, pharmaceuticals, plastics and perfluorinated substances among others. However, using concentrations measured along waterways to identify sources of contaminants and predict their fate is complicated by downstream mixing. In this study, spot measurements of ecotoxicologically relevant chemicals collected along the Wandle, a rare urban chalk stream that flows into south London, UK, are combined with drainage network topology to objectively calculate locations and concentrations of contaminant sources using an inverse modelling approach. Mixing is assumed to be conservative, and the location of sources and their concentrations are treated as unknowns to be identified. Calculated source concentrations of thirteen chemicals, which range from below detection limit (a few ng/l) up to 1000 ng/l, are used to predict concentrations of chemicals downstream. Contaminant fluxes are estimated by combining results with flow gauge measurements. Predicted concentrations and estimates of probable no-effect values indicate that chemical risk quotients are high for insecticides imidacloprid and acetamiprid, and above negligible for the pharmaceutical diclofenac among others. Principal component analysis revealed signatures of two distinct chemical mixtures. First, pharmaceuticals and insecticides were associated with a subcatchment containing a known point source of treated wastewater---the Beddington wastewater treatment plant. Second, illicit drugs and salicylic acid are associated with multiple sources, interpreted as markers of input from untreated sewage including CSOs, misconnections, runoff, and direct disposal throughout the catchment. Finally, a simple algorithmic approach that incorporates network topology is developed to design sampling campaigns to improve resolution of source apportionment. Inverse modelling of contaminant measurements could provide objective means to apportion sources in waterways from spot samples in catchments on a large scale.